#434017

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

963 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,11,16,12,17,10,27)(7,18,21,30,9,15,20,13)(8,22,25,33,26,14,28,34)(19,31,23,36,24,32,35,29);(5,7,10,20,12,9,11,21)(6,13,16,18,17,30,27,15)(8,23,25,24,26,35,28,19)(14,31,33,29,22,32,34,36);(5,8,10,28,12,26,11,25)(6,14,16,34,17,22,27,33)(7,19,21,23,9,24,20,35)(13,29,15,32,30,36,18,31);(2,4)(5,9)(6,15)(7,12)(8,24)(10,20)(11,21)(13,27)(14,32)(16,30)(17,18)(19,26)(22,31)(23,25)(28,35)(29,33)(34,36)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341213);